US-France Cooperative Research: Conservation Laws in Continuum Mechanics

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This award will support collaborative research between Dr. Marshall Slemrod, University of Wisconsin and two French applied mathe- maticians: Professor Michel Rascle, University of Nice and Professor Denis Serre, Ecole Normale Superieure de Lyon. The objective of the project is to study the existance of solutions to systems of conservation laws arising in continuum mechanics. In particular, the investigators will study the decay, propagation and creation of oscillations using the tools of compensated compactness and the Young measure. One of the most important branches of partial differential equations is the area of conservation laws. In fact most mechanical systems are described by such equations, e.g. motion of elastic bodies, gases fluids, plasmas, etc. Hence their study both analytically and numerically is exceptionally important. In the proposed project, the investigators will focus on the following: 1) dynamics of the Young measure associated with a sequence of approximate solutions 2) systems of conservation laws describing phase transitions. Dr. Slemrod and his French colleagues are leading researchers in a small group of mathematicians working on the analysis of physically reasonable problems in conservation laws. Solution of the proposed problems would be a significant step toward the understanding of one of the central problems of modern applied mathematics: the dynamics of discontinuous, oscillatory, measure-valued physical systems.